Recent zbMATH articles in MSC 35Lhttps://www.zbmath.org/atom/cc/35L2021-06-15T18:09:00+00:00WerkzeugFinite lifespan of solutions of the semilinear wave equation in the Einstein-de Sitter spacetime.https://www.zbmath.org/1460.352212021-06-15T18:09:00+00:00"Galstian, Anahit"https://www.zbmath.org/authors/?q=ai:galstian.anahit"Yagdjian, Karen"https://www.zbmath.org/authors/?q=ai:yagdjian.karenIn the present paper the authors study a suitable class of weighted initial value problems for wave type models with friction and source nonlinearities of power type (gauge invariant and non gauge invariant form as well). There appear singular time-dependent coefficients (at \(t=0\)) in the elliptic part and the damping term as well. This singular behavior motivates to pose weighted initial conditions instead of classical Cauchy conditions.
The elliptic operator is modulo a factor in self-adjoint form (coefficients are smooth and depend on the spatial variables only).
Outside of some compact set the elliptic operator is the Laplace operator times a positive constant.
The authors study blow-up phenomena of local (in time) Sobolev solutions even for small data. They discuss this matter for the singular
model and for the regular model after shifting the initial plane. In this way they verify that the blow-up behavior is caused by the nonlinear term on the right-hand side. They propose some critical value for exponents in the power nonlinearities.
Some applications to models from cosmology are given.
The proofs of the main result use Kato lemma for suitable ordinary differential inequalities, some explicit representations of solutions, operator identities and Liouville-type transformations.
Reviewer: Michael Reissig (Freiberg)Interaction of delta shock waves for a nonsymmetric Keyfitz-Kranzer system of conservation laws.https://www.zbmath.org/1460.352272021-06-15T18:09:00+00:00"de la Cruz, Richard"https://www.zbmath.org/authors/?q=ai:de-la-cruz.richard"Santos, Marcelo"https://www.zbmath.org/authors/?q=ai:santos.marcelo-m"Abreu, Eduardo"https://www.zbmath.org/authors/?q=ai:abreu.eduardoSummary: In this work, the mechanism for the formation of the delta shock wave is analyzed to deal with interaction of delta shock waves and contact discontinuities for a system of Keyfitz-Kranzer type by means of analysis and solutions of Riemann problems. A set of numerical experiments are provided, illustrating the theoretical findings numerically. A brief survey of the Keyfitz-Kranzer systems as a base model of fundamental nonlinear phenomena in applications is provided aiming to shed light on the intricate wave structure for other related models of conservation laws appearing in applied sciences.Uniqueness of dissipative solutions to the complete Euler system.https://www.zbmath.org/1460.352712021-06-15T18:09:00+00:00"Ghoshal, Shyam Sundar"https://www.zbmath.org/authors/?q=ai:ghoshal.shyam-sundar"Jana, Animesh"https://www.zbmath.org/authors/?q=ai:jana.animeshSummary: Dissipative solutions have recently been studied as a generalized concept for weak solutions of the complete Euler system. Apparently, these are expectations of suitable measure valued solutions. Motivated from \textit{E. Feireisl} et al. [Commun. Partial Differ. Equations 44, No. 12, 1285--1298 (2019; Zbl 1428.35325)], we impose a one-sided Lipschitz bound on velocity component as uniqueness criteria for a weak solution in Besov space \(B^{\alpha ,\infty}_p\) with \(\alpha >1/2\). We prove that the Besov solution satisfying the above mentioned condition is unique in the class of dissipative solutions. In the later part of this article, we prove that the one sided Lipschitz condition gives uniqueness among weak solutions with the Besov regularity, \(B^{\alpha,\infty}_3\) for \(\alpha >1/3\). Our proof relies on commutator estimates for Besov functions and the relative entropy method.Representations of the solutions of the first-order elliptic and hyperbolic systems via harmonic and wave functions Respectively.https://www.zbmath.org/1460.350642021-06-15T18:09:00+00:00"Tokibetov, J."https://www.zbmath.org/authors/?q=ai:tokibetov.j-a"Abduakhitova, G."https://www.zbmath.org/authors/?q=ai:abduakhitova.g-e"Assadi, A."https://www.zbmath.org/authors/?q=ai:assadi.amir-hSummary: In this paper representations of the solutions of all first-order elliptic and hyperbolic systems in three-dimensional and four-dimensional spaces are obtained through the derivatives of harmonic and wave functions, respectively.Uniform stabilization of the Klein-Gordon system.https://www.zbmath.org/1460.352242021-06-15T18:09:00+00:00"Cavalcanti, Marcelo M."https://www.zbmath.org/authors/?q=ai:cavalcanti.marcelo-moreira"Delatorre, Leonel G."https://www.zbmath.org/authors/?q=ai:delatorre.leonel-g"Soares, Daiane C."https://www.zbmath.org/authors/?q=ai:soares.daiane-c"Martinez, Victor Hugo Gonzalez"https://www.zbmath.org/authors/?q=ai:gonzalez-martinez.victor-hugo"Zanchetta, Janaina P."https://www.zbmath.org/authors/?q=ai:zanchetta.janaina-pedrosoSummary: We consider the Klein-Gordon system posed in an inhomogeneous medium \(\Omega\) with smooth boundary \(\partial\Omega\) subject to two localized dampings. The first one is of the type viscoelastic and is distributed around a neighborhood \(\omega\) of the boundary according to the Geometric Control Condition. The second one is a frictional damping and we consider it hurting the geometric condition of control. We show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space. For this purpose, refined microlocal analysis arguments are considered by exploiting ideas due to \textit{N. Burq} and \textit{P. Gérard} [Contrôle optimal des équations aux dérivées partielles. Paris: Ecole Polytechnique (2002), \url{https://www.imo.universite-paris-saclay.fr/~pgerard/coursX.pdf}]. Although the present problem has some similarity to the reference [\textit{M. M. Cavalcanti} et al., J. Differ. Equations 268, No. 2, 447--489 (2020; Zbl 1429.35156)] it is important to mention that due to the Kelvin-Voigt dissipation character associated with the nonlinearity of the problem the approach used is completely new, which is the main purpose of this paper.Nonlocal and nonlinear effects in hyperbolic heat transfer in a two-temperature model.https://www.zbmath.org/1460.800012021-06-15T18:09:00+00:00"Sellitto, A."https://www.zbmath.org/authors/?q=ai:sellitto.antonio"Carlomagno, I."https://www.zbmath.org/authors/?q=ai:carlomagno.isabella"Di Domenico, M."https://www.zbmath.org/authors/?q=ai:di-domenico.maria-carlaManufacturing of micro- and nanodevices makes it necessary to analyze heat transfer on nanoscale. It is impossible to use classical Fourier law in this case because characteristic length scale of the problem is comparable with mean free path of heat carreers. For example, these difficulties take place while modelling of electrons and phonons passing through a crystal lattice.
In order to describe the physical system mentioned above, the authors use a two-temperature model that takes into account nonlocal and nonlinear effects as well. In particular, the authors introduce relaxation times and mean free paths of electrons and phonons as important model parameters in addition to the ``classical'' parameters such as specific heats and heat conductivities.
Though this model is not obtained by rigorous microscopic derivation, the authors discuss its consistency with statistics of Bose-Einstein and Fermi-Dirac as well as with the 2nd law of thermodynamics.
The techniques of acceleration waves is then used to describe the heat propagation in the two-temperature medium. The dependence of heat wave speeds and amplitudes in electronic and phononic gases on the physical system characteristics is found. In particular, conditions are derived that show when the waves under discussion become shock waves.
Reviewer: Aleksey Syromyasov (Saransk)Sampling in thermoacoustic tomography.https://www.zbmath.org/1460.354052021-06-15T18:09:00+00:00"Mathison, Chase"https://www.zbmath.org/authors/?q=ai:mathison.chaseSummary: We explore the effect of sampling rates when measuring data given by \(Mf\) for special operators \(M\) arising in Thermoacoustic Tomography. We start with sampling requirements on \(Mf\) given \(f\) satisfying certain conditions. After this we discuss the resolution limit on \(f\) posed by the sampling rate of \(Mf\) without assuming any conditions on these sampling rates. Next we discuss aliasing artifacts when \(Mf\) is known to be under sampled in one or more of its variables. Finally, we discuss averaging of measurement data and resulting aliasing and artifacts, along with a scheme for anti-aliasing.Longtime dynamics for a type of suspension bridge equation with past history and time delay.https://www.zbmath.org/1460.350392021-06-15T18:09:00+00:00"Liu, Gongwei"https://www.zbmath.org/authors/?q=ai:liu.gongwei"Feng, Baowei"https://www.zbmath.org/authors/?q=ai:feng.baowei"Yang, Xinguang"https://www.zbmath.org/authors/?q=ai:yang.xinguangSummary: In this paper, we investigate a suspension bridge equation with past history and time delay effects, defined in a bounded domain \(\Omega\) of \(\mathbb{R}^N\). Many researchers have considered the well-posedness, energy decay of solution and existence of global attractors for suspension bridge equation without memory or delay. But as far as we know, there are no results on the suspension bridge equation with both memory and time delay. The purpose of this paper is to show the existence of a global attractor which has finite fractal dimension by using the methods developed by Chueshov and Lasiecka. Result on exponential attractors is also proved. We also establish the exponential stability under some conditions. These results are extension and improvement of earlier results.Quadratic optimal control for bilinear systems.https://www.zbmath.org/1460.352372021-06-15T18:09:00+00:00"Yahyaoui, Soufiane"https://www.zbmath.org/authors/?q=ai:yahyaoui.soufiane"Ouzahra, Mohamed"https://www.zbmath.org/authors/?q=ai:ouzahra.mohamedSummary: In this work, we will investigate the quadratic optimal control for bilinear systems. We will first study the existence of a solution for the considered optimal control. Then, we will focus on a special class of bilinear systems for which the quadratic optimal control can be expressed in a feedback law form. The approach relies on the conditions of optimality and linear semi-group theory.
For the entire collection see [Zbl 1459.35003].Conservation and constitutive equations in curvilinear coordinates.https://www.zbmath.org/1460.352262021-06-15T18:09:00+00:00"Cossali, Gianpietro Elvio"https://www.zbmath.org/authors/?q=ai:cossali.gianpietro-elvio"Tonini, Simona"https://www.zbmath.org/authors/?q=ai:tonini.simonaSummary: The formulation of the conservation and constitutive differential equations derived in the previous chapters was obtained under the implicit assumption that the coordinate system was a Cartesian one. In practical problems it is sometime useful to switch to more natural coordinate systems, where the actual form of the differential equations may be simplified, thanks to some symmetry properties of the problem. For example, when dealing with the heating and evaporation of a spherical drop, the natural coordinate system is the spherical one, since in such a system the governing differential equations may assume a much simpler form.
For the entire collection see [Zbl 1459.94002].Riemann problem and wave interactions for a class of strictly hyperbolic systems of conservation laws.https://www.zbmath.org/1460.352312021-06-15T18:09:00+00:00"Zhang, Yu"https://www.zbmath.org/authors/?q=ai:zhang.yu.3"Zhang, Yanyan"https://www.zbmath.org/authors/?q=ai:zhang.yanyanSummary: A class of strictly hyperbolic systems of conservation laws are proposed and studied. Firstly, the Riemann problem with initial data of two piecewise constant states is constructively solved. The solutions involving contact discontinuities and delta shock waves are obtained. The generalized Rankine-Hugoniot relation and entropy condition for the delta shock wave are clarified and the existence and uniqueness of the delta-shock solution is proved. Furthermore, the global structure of solutions with five different configurations is constructed via investigating the interactions of delta shock waves and contact discontinuities. Finally, we present a typical example to illustrate the application of the system introduced.Global stability of some totally geodesic wave maps.https://www.zbmath.org/1460.352332021-06-15T18:09:00+00:00"Abbrescia, Leonardo Enrique"https://www.zbmath.org/authors/?q=ai:abbrescia.leonardo-enrique"Chen, Yuan"https://www.zbmath.org/authors/?q=ai:chen.yuanSummary: We prove that wave maps that factor as \(\mathbb{R}^{1 + d} \xrightarrow{\varphi_\mathrm{S}} \mathbb{R} \xrightarrow{\varphi_1} M\), subject to a sign condition, are globally nonlinear stable under small compactly supported perturbations when \(M\) is a spaceform. The main innovation is our assumption on \(\varphi_{\mathrm{S}} \), namely that it be a semi-Riemannian submersion. This implies that the background solution has infinite total energy, making this, to the best of our knowledge, the first stability result for factored wave maps with infinite energy backgrounds. We prove that the equations of motion for the perturbation decouple into a nonlinear wave-Klein-Gordon system. We prove global existence for this system and improve on the known regularity assumptions for equations of this type.On the Strauss index of semilinear Tricomi equation.https://www.zbmath.org/1460.352352021-06-15T18:09:00+00:00"He, Daoyin"https://www.zbmath.org/authors/?q=ai:he.daoyin"Witt, Ingo"https://www.zbmath.org/authors/?q=ai:witt.ingo"Yin, Huicheng"https://www.zbmath.org/authors/?q=ai:yin.huichengSummary: In our previous papers, we have given a systematic study on the global existence versus blowup problem for the small-data solution \(u\) of the multi-dimensional semilinear Tricomi equation
\[
\partial_t^2u-t\Delta u=|u|^p,\quad \big(u(0,\cdot),\partial_t u(0,\cdot)\big)=(u_0, u_1),
\]
where \(t>0\), \(x\in\mathbb{R}^n\), \(n\geq 2\), \(p>1\), and \(u_i\in C_0^{\infty}(\mathbb{R}^n)\) \((i=0,1)\). In this article, we deal with the remaining 1-D problem, for which the stationary phase method for multi-dimensional case fails to work and the large time decay rate of \(\|u(t, \cdot)\|_{L^\infty_x(\mathbb{R})}\) is not enough. The main ingredient of the proof in this paper is to use the structure of the linear equation to get the suitable decay rate of \(u\) in \(t\), then the crucial weighted Strichartz estimates are established and the global existence of solution \(u\) is proved when \(p>5\).Non-uniqueness of delta shocks and contact discontinuities in the multi-dimensional model of Chaplygin gas.https://www.zbmath.org/1460.352692021-06-15T18:09:00+00:00"Březina, Jan"https://www.zbmath.org/authors/?q=ai:brezina.jan"Kreml, Ondřej"https://www.zbmath.org/authors/?q=ai:kreml.ondrej"Mácha, Václav"https://www.zbmath.org/authors/?q=ai:macha.vaclavSummary: We study the Riemann problem for the isentropic compressible Euler equations in two space dimensions with the pressure law describing the Chaplygin gas. It is well known that there are Riemann initial data for which the 1D Riemann problem does not have a classical \textit{BV} solution, instead a \(\delta\)-shock appears, which can be viewed as a generalized measure-valued solution with a concentration measure in the density component. We prove that in the case of two space dimensions there exist infinitely many bounded admissible weak solutions starting from the same initial data. Moreover, we show the same property also for a subset of initial data for which the classical 1D Riemann solution consists of two contact discontinuities. As a consequence of the latter result we observe that any criterion based on the principle of maximal dissipation of energy will not pick the classical 1D solution as the physical one. In particular, not only the criterion based on comparing dissipation rates of total energy but also a stronger version based on comparing dissipation measures fails to pick the 1D solution.Entropy supplementary conservation law for non-linear systems of PDEs with non-conservative terms: application to the modelling and analysis of complex fluid flows using computer algebra.https://www.zbmath.org/1460.352252021-06-15T18:09:00+00:00"Cordesse, Pierre"https://www.zbmath.org/authors/?q=ai:cordesse.pierre"Massot, Marc"https://www.zbmath.org/authors/?q=ai:massot.marcSummary: In the present contribution, we investigate first-order nonlinear systems of partial differential equations which are constituted of two parts: a system of conservation laws and nonconservative first-order terms. Whereas the theory of first-order systems of conservation laws is well established and the conditions for the existence of supplementary conservation laws, and more specifically of an entropy supplementary conservation law for smooth solutions, well known, there exists so far no general extension to obtain such supplementary conservation laws when non-conservative terms are present. We propose a framework in order to extend the existing theory and show that the presence of non-conservative terms somewhat complexifies the problem since numerous combinations of the conservative and non-conservative terms can lead to a supplementary conservation law. We then identify a restricted framework in order to design and analyze physical models of complex fluid flows by means of computer algebra and thus obtain the entire ensemble of possible combination of conservative and non-conservative terms with the objective of obtaining specifically an entropy supplementary conservation law. The theory as well as developed computer algebra tool are then applied to a Baer-Nunziato two-phase flow model and to a multicomponent plasma fluid model. The first one is a first-order fluid model, with non-conservative terms impacting on the linearly degenerate field and requires a closure since there is no way to derive interfacial quantities from averaging principles and we need guidance in order to close the pressure and velocity of the interface and the thermodynamics of the mixture. The second one involves first-order terms for the heavy species coupled to second-order terms for the electrons, the non-conservative terms impact the genuinely nonlinear fields and the model can be rigorously derived from kinetic theory. We show how the theory allows to recover the whole spectrum of closures obtained so far in the literature for the two-phase flow system as well as conditions when one aims at extending the thermodynamics and also applies to the plasma case, where we recover the usual entropy supplementary equation, thus assessing the effectiveness and scope of the proposed theory.The Boussinesq system revisited.https://www.zbmath.org/1460.352952021-06-15T18:09:00+00:00"Molinet, Luc"https://www.zbmath.org/authors/?q=ai:molinet.luc"Talhouk, Raafat"https://www.zbmath.org/authors/?q=ai:talhouk.raafat"Zaiter, Ibtissam"https://www.zbmath.org/authors/?q=ai:zaiter.ibtissamOn a class of degenerate abstract parabolic problems and applications to some eddy current models.https://www.zbmath.org/1460.353392021-06-15T18:09:00+00:00"Pauly, Dirk"https://www.zbmath.org/authors/?q=ai:pauly.dirk"Picard, Rainer"https://www.zbmath.org/authors/?q=ai:picard.rainer-h"Trostorff, Sascha"https://www.zbmath.org/authors/?q=ai:trostorff.sascha"Waurick, Marcus"https://www.zbmath.org/authors/?q=ai:waurick.marcusThe authors develop a framework for parabolic problems which can be degenerate in certain spatial regions. The approach used by the authors is related to evolution equations in Hilbert spaces, and involves only minimal assumptions on the boundary. This framework is used to analyze the structure of the degenerate eddy current problem. This eddy current problem is then justified as a limiting model of Maxwell's equations.
Reviewer: Eric Stachura (Marietta)Representation of solutions of the Cauchy problem for a one-dimensional Schrödinger equation with a smooth bounded potential by quasi-Feynman formulae.https://www.zbmath.org/1460.810172021-06-15T18:09:00+00:00"Grishin, Denis V."https://www.zbmath.org/authors/?q=ai:grishin.denis-v"Pavlovskiy, Yan Yu."https://www.zbmath.org/authors/?q=ai:pavlovskiy.yan-yuThe global supersonic flow with vacuum state in a 2D convex duct.https://www.zbmath.org/1460.352722021-06-15T18:09:00+00:00"Li, Jintao"https://www.zbmath.org/authors/?q=ai:li.jintao"Shen, Jindou"https://www.zbmath.org/authors/?q=ai:shen.jindou"Xu, Gang"https://www.zbmath.org/authors/?q=ai:xu.gangSummary: This paper concerns the motion of the supersonic potential flow in a two-dimensional expanding duct. In the case that two Riemann invariants are both monotonically increasing along the inlet, which means the gases are spread at the inlet, we obtain the global solution by solving the problem in those inner and border regions divided by two characteristics in \((x, y)\)-plane, and the vacuum will appear in some finite place adjacent to the boundary of the duct. In addition, we point out that the vacuum here is not the so-called physical vacuum. On the other hand, for the case that at least one Riemann invariant is strictly monotonic decreasing along some part of the inlet, which means the gases have some local squeezed properties at the inlet, we show that the \(C^1\) solution to the problem will blow up at some finite location in the non-convex duct.A second order fractional differential equation under effects of a super damping.https://www.zbmath.org/1460.353702021-06-15T18:09:00+00:00"Charão, Ruy Coimbra"https://www.zbmath.org/authors/?q=ai:charao.ruy-coimbra"Espinoza, Juan Torres"https://www.zbmath.org/authors/?q=ai:espinoza.juan-torres"Ikehata, Ryo"https://www.zbmath.org/authors/?q=ai:ikehata.ryoSummary: In this work we study asymptotic properties of global solutions for an initial value problem of a second order fractional differential equation with structural damping. The evolution equation considered includes plate equation problems. We show asymptotic profiles depending on the exponents of the Laplace operators involved in the equation and optimality of the decay rates for the associated energy and the \(L^2\) norm of solutions.\( \sigma \)-evolution models with low regular time-dependent effective structural damping.https://www.zbmath.org/1460.350432021-06-15T18:09:00+00:00"Vargas Junior, Edson Cilos"https://www.zbmath.org/authors/?q=ai:vargas.edson-cilos-jun"da Luz, Cleverson Roberto"https://www.zbmath.org/authors/?q=ai:da-luz.cleverson-robertoSummary: In this work we investigate decay rates for Sobolev solutions of \(\sigma \)-evolution equations in \(\mathbb{R}^n\) under effects of a damping term represented by the action of a fractional Laplacian operator and a time-dependent coefficient, \(b(t) ( - \Delta )^\theta u_t\). We assume the coefficient \(b = b(t)\) is of low regularity with the shape of a regular function \(g\), that is, \( a_1 b(t) \leq g(t) \leq a_2 b(t)\) for large times. For this purpose, we developed a new technique to handle the lack of control of the derivative of \(b = b(t)\).Fractional Landweber method for an initial inverse problem for time-fractional wave equations.https://www.zbmath.org/1460.353752021-06-15T18:09:00+00:00"Huynh, Le Nhat"https://www.zbmath.org/authors/?q=ai:huynh.le-nhat"Zhou, Yong"https://www.zbmath.org/authors/?q=ai:zhou.yong|zhou.yong.1"O'Regan, Donal"https://www.zbmath.org/authors/?q=ai:oregan.donal"Tuan, Nguyen Huy"https://www.zbmath.org/authors/?q=ai:nguyen-huy-tuan.Summary: In this paper, we consider the initial inverse problem (backward problem) for an inhomogeneous time-fractional wave equation in a general bounded domain. We show that the backward problem is ill-posed, and we propose a regularizing scheme using a fractional Landweber regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules.A new numerical method for level set motion in normal direction used in optical flow estimation.https://www.zbmath.org/1460.650172021-06-15T18:09:00+00:00"Frolkovič, Peter"https://www.zbmath.org/authors/?q=ai:frolkovic.peter"Kleinová, Viera"https://www.zbmath.org/authors/?q=ai:kleinova.vieraSummary: We present a new numerical method for the solution of level set advection equation describing a motion in normal direction for which the speed is given by the sign function of the difference of two given functions. Taking one function as the initial condition, the solution evolves towards the second given function. One of possible applications is an optical flow estimation to find a deformation between two images in a video sequence. The new numerical method is based on a bilinear interpolation of discrete values as used for the representation of images. Under natural assumptions, it ensures a monotone decrease of the absolute difference between the numerical solution and the target function, and it handles properly the discontinuity in the speed due to the dependence on the sign function. To find the deformation between two functions (or images), the backward tracking of characteristics is used. Two numerical experiments are presented, one with an exact solution to show an experimental order of convergence and one based on two images of lungs to illustrate a possible application of the method for the optical flow estimation.Strichartz estimates and Fourier restriction theorems on the Heisenberg group.https://www.zbmath.org/1460.353652021-06-15T18:09:00+00:00"Bahouri, Hajer"https://www.zbmath.org/authors/?q=ai:bahouri.hajer"Barilari, Davide"https://www.zbmath.org/authors/?q=ai:barilari.davide"Gallagher, Isabelle"https://www.zbmath.org/authors/?q=ai:gallagher.isabelleSummary: This paper is dedicated to the proof of Strichartz estimates on the Heisenberg group \(\mathbb{H}^d\) for the linear Schrödinger and wave equations involving the sublaplacian. The Schrödinger equation on \(\mathbb{H}^d\) is an example of a totally non-dispersive evolution equation: for this reason the classical approach that permits to obtain Strichartz estimates from dispersive estimates is not available. Our approach, inspired by the Fourier transform restriction method initiated in [\textit{P. A. Tomas}, Bull. Am. Math. Soc. 81, 477--478 (1975; Zbl 0298.42011)], is based on Fourier restriction theorems on \(\mathbb{H}^d\), using the non-commutative Fourier transform on the Heisenberg group. It enables us to obtain also an anisotropic Strichartz estimate for the wave equation, for a larger range of indices than was previously known.Observability and stabilization of \(1-D\) wave equations with moving boundary feedback.https://www.zbmath.org/1460.352202021-06-15T18:09:00+00:00"Lu, Liqing"https://www.zbmath.org/authors/?q=ai:lu.liqing"Feng, Yating"https://www.zbmath.org/authors/?q=ai:feng.yatingSummary: In this paper, we are concerned with a wave equation on a time-dependent domain with a Dirichlet boundary condition at the endpoint \(x=0\) and a boundary feedback at the moving endpoint \(x=kt\). We discuss the stabilization and exact boundary observability of the 1-dimensional wave equation with moving boundary feedback. By using generalized Fourier series and Parseval's equality in weighted \(L^2-\) spaces, we derive a precise polynomial asymptotic stability for the energy function of the solution. Moreover, the exact boundary observabilities of the solution are established in minimal time. The observability constants are explicitly given at each endpoint.High-order finite volume WENO schemes for non-local multi-class traffic flow models.https://www.zbmath.org/1460.651062021-06-15T18:09:00+00:00"Chiarello, Felisia A."https://www.zbmath.org/authors/?q=ai:chiarello.felisia-angela"Goatin, Paola"https://www.zbmath.org/authors/?q=ai:goatin.paola"Villada, Luis M."https://www.zbmath.org/authors/?q=ai:villada.luis-miguelSummary: This paper focuses on the numerical approximation of a class of non-local systems of conservation laws in one space dimension, arising in traffic modeling, proposed by \textit{F. A. Chiarello} and \textit{P. Goatin} [Netw. Heterog. Media 14, No. 2, 371--387 (2019; Zbl 1426.35153)]. We present the multi-class version of the Finite Volume WENO (FV-WENO) schemes \textit{C. Chalons} et al. [SIAM J. Sci. Comput. 40, No. 1, A288--A305 (2018; Zbl 1387.35406)], with quadratic polynomial reconstruction in each cell to evaluate the non-local terms in order to obtain high-order of accuracy. Simulations using FV-WENO schemes for a multi-class model for autonomous and human-driven traffic flow are presented for \(M=3\).
For the entire collection see [Zbl 1453.35003].Convergence rate from hyperbolic systems of balance laws to parabolic systems.https://www.zbmath.org/1460.350152021-06-15T18:09:00+00:00"Li, Yachun"https://www.zbmath.org/authors/?q=ai:li.yachun"Peng, Yue-Jun"https://www.zbmath.org/authors/?q=ai:peng.yuejun"Zhao, Liang"https://www.zbmath.org/authors/?q=ai:zhao.liang.2|zhao.liang.4|zhao.liang.1|zhao.liang.5|zhao.liang.3|zhao.liangSummary: It is proved recently that partially dissipative hyperbolic systems converge globally-in-time to parabolic systems in a slow time scaling, when initial data are smooth and sufficiently close to constant equilibrium states. Based on this result, we establish error estimates between the smooth solutions of the hyperbolic systems of balance laws and those of the parabolic limit systems in one space dimension. The proof of the error estimates uses a stream function technique together with energy estimates. As applications of the results, we give five examples arising from physical models.Stability of non-constant equilibrium solutions for the full compressible Navier-Stokes-Maxwell system.https://www.zbmath.org/1460.352822021-06-15T18:09:00+00:00"Feng, Yue-Hong"https://www.zbmath.org/authors/?q=ai:feng.yuehong"Li, Xin"https://www.zbmath.org/authors/?q=ai:li.xin.9|li.xin.3|li.xin.7|li.xin.10|li.xin.6|li.xin.4|li.xin.2|li.xin.1|li.xin.12|li.xin.13|li.xin.5|li.xin.15|li.xin|li.xin.11|li.xin.14"Wang, Shu"https://www.zbmath.org/authors/?q=ai:wang.shuSummary: In this article we consider a Cauchy problem for the full compressible Navier-Stokes-Maxwell system arising from viscosity plasmas. This system is quasilinear hyperbolic-parabolic. With the help of techniques of symmetrizers and the smallness of non-constant equilibrium solutions, we establish that global smooth solutions exist and converge to the equilibrium solution as the time approaches infinity. This result is obtained for initial data close to the steady-states. As a byproduct, we obtain the global stability of solutions near the equilibrium states for the full compressible Navier-Stokes-Poisson system in a three-dimensional torus.A note on the slow convergence of solutions to conservation laws with mean curvature diffusions.https://www.zbmath.org/1460.350422021-06-15T18:09:00+00:00"Strani, Marta"https://www.zbmath.org/authors/?q=ai:strani.martaSummary: We study the asymptotic behaviour of solutions to a scalar conservation law with a mean curvature's type diffusion, focusing our attention to the stability/metastability properties of the steady state. In particular, we show the existence of a unique steady state that slowly converges to its asymptotic configuration, with a speed rate which is exponentially small with respect to the viscosity parameter \(\varepsilon\); the rigorous results are also validated by numerical simulations.Global entropy solutions to the compressible Euler equations in the isentropic nozzle flow.https://www.zbmath.org/1460.350682021-06-15T18:09:00+00:00"Tsuge, Naoki"https://www.zbmath.org/authors/?q=ai:tsuge.naokiThe auhtor studies the one-dimensional non-steady isentropic compressible Euler flow in a nozzle. The nozzle is infinitely long and described by an \(x\)-dependent cross-section function in the equations. Equations are written in terms of density and momentum. The gas is barotropic, so that pressure is a power function of density with the adiabatic exponent is from [1,5/3]. The Cauchy problem for arbitrarily large initial data is studied. The aim is to establish the global existence of the entropy solution with sonic state.
For the entire collection see [Zbl 1453.35003].
Reviewer: Ilya A. Chernov (Petrozavodsk)Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity.https://www.zbmath.org/1460.352362021-06-15T18:09:00+00:00"Han, Zheng"https://www.zbmath.org/authors/?q=ai:han.zheng"Fang, Daoyuan"https://www.zbmath.org/authors/?q=ai:fang.daoyuanSummary: We prove an almost global existence result for the Klein-Gordon equation with the Kirchhoff-type nonlinearity on \(\mathbb{T}^d\) with Cauchy data of small amplitude \(\epsilon\). We show a lower bound \(\epsilon^{-2N-2}\) for the existence time with any natural number \(N\). The proof relies on the method of normal forms and induction. The structure of the nonlinearity is good enough that proceeds normal forms up to any order.Nonexistence for hyperbolic problems on Riemannian manifolds.https://www.zbmath.org/1460.353642021-06-15T18:09:00+00:00"Monticelli, Dario D."https://www.zbmath.org/authors/?q=ai:monticelli.dario-daniele"Punzo, Fabio"https://www.zbmath.org/authors/?q=ai:punzo.fabio"Squassina, Marco"https://www.zbmath.org/authors/?q=ai:squassina.marcoSummary: We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems defined on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on the whole Euclidean space. A general weight function depending on spacetime is allowed in front of the power nonlinearity.Restriction inequalities for the hyperbolic hyperboloid.https://www.zbmath.org/1460.420102021-06-15T18:09:00+00:00"Bruce, Benjamin Baker"https://www.zbmath.org/authors/?q=ai:bruce.benjamin-baker"Oliveira e. Silva, Diogo"https://www.zbmath.org/authors/?q=ai:oliveira-e-silva.diogo"Stovall, Betsy"https://www.zbmath.org/authors/?q=ai:stovall.betsySummary: In this article we establish new inequalities, both conditional and unconditional, for the restriction problem associated to the hyperbolic, or one-sheeted, hyperboloid in three dimensions, endowed with a Lorentz-invariant measure. These inequalities are unconditional (and optimal) in the bilinear range \(q>\frac{10}{3}\).General decay result of solutions for viscoelastic wave equation with Balakrishnan-Taylor damping and a delay term.https://www.zbmath.org/1460.350372021-06-15T18:09:00+00:00"Gheraibia, Billel"https://www.zbmath.org/authors/?q=ai:gheraibia.billel"Boumaza, Nouri"https://www.zbmath.org/authors/?q=ai:boumaza.nouriThe authors study the initial-boundary value problem
\begin{align*}
&\partial_{tt}u -\left(u+ b\|\nabla u\|^2_2+\alpha \int_\Omega\nabla u\cdot\nabla u_t\,\mathrm{d}x \right)\Delta u + \sigma(t)\int_0^t g(t-s)\delta u(s)\,\mathrm{d}s\\
&+\mu_1|u_t|^{m-2}u_t+\mu_2|u_t(t-\tau)|^{m-2}u_t(t-\tau)=0 \ \text{in}\ \Omega\times (0,\infty),\\
&u(x,t)=0\ \text{on}\ \partial\Omega\times (0,\infty),\quad u(x,0)=u_0\ \text{in}\ \Omega,\\
& u_t(x,t-\tau)=f_0(x,t-\tau)\ \text{in}\ \Omega\times (0,\tau),
\end{align*}
where \(\Omega\) is a bounded Lipschitz domain in \(\mathbb{R}^n,\ m\ge 2,\ \tau>0\) is a time delay, \(\sigma\) and \(g\) are positive functions.
The considered problem represents one of a lot various generalizations of similar problems. Novelty here is the term \(|u_t|^{m-2}u_t\) and its delay instead of \(u_t\) and its delay. Local existence and uniqueness theorem is stated without any proof. The main result is the decay of the energy in a form \[E(t)\le Ke^{-k\int_0^t\zeta(s)\sigma(s)ds}\ \forall t\ge t_1>0.\]
Reviewer: Igor Bock (Bratislava)SBV regularity for Burgers-Poisson equation.https://www.zbmath.org/1460.352282021-06-15T18:09:00+00:00"Gilmore, Steven"https://www.zbmath.org/authors/?q=ai:gilmore.steven"Nguyen, Khai T."https://www.zbmath.org/authors/?q=ai:nguyen.khai-tSummary: The SBV regularity of weak entropy solutions to the Burgers-Poisson equation for initial data in \(\mathbf{L}^1(\mathbb{R})\) is considered. We show that the derivative of a solution consists of only the absolutely continuous part and the jump part.On the wave equation with multiplicities and space-dependent irregular coefficients.https://www.zbmath.org/1460.352192021-06-15T18:09:00+00:00"Garetto, Claudia"https://www.zbmath.org/authors/?q=ai:garetto.claudiaSummary: In this paper we study the well-posedness of the Cauchy problem for a wave equation with multiplicities and space-dependent irregular coefficients. As in [\textit{C. Garetto} and \textit{M. Ruzhansky}, Arch. Ration. Mech. Anal. 217, No. 1, 113--154 (2015; Zbl 1320.35181)], in order to give a meaningful notion of solution, we employ the notion of very weak solution, which construction is based on a parameter dependent regularisation of the coefficients via mollifiers. We prove that, even with distributional coefficients, a very weak solution exists for our Cauchy problem and it converges to the classical one when the coefficients are smooth. The dependence on the mollifiers of very weak solutions is investigated at the end of the paper in some instructive examples.Infinite energy solutions for weakly damped quintic wave equations in \(\mathbb{R}^3\).https://www.zbmath.org/1460.350402021-06-15T18:09:00+00:00"Mei, Xinyu"https://www.zbmath.org/authors/?q=ai:mei.xinyu"Savostianov, Anton"https://www.zbmath.org/authors/?q=ai:savostianov.anton-k|savostianov.anton"Sun, Chunyou"https://www.zbmath.org/authors/?q=ai:sun.chunyou"Zelik, Sergey"https://www.zbmath.org/authors/?q=ai:zelik.sergey-vSummary: The paper gives a comprehensive study of infinite-energy solutions and their long-time behavior for semi-linear weakly damped wave equations in \(\mathbb{R}^3\) with quintic nonlinearities. This study includes global well-posedness of the so-called Shatah-Struwe solutions, their dissipativity, the existence of a locally compact global attractors (in the uniformly local phase spaces) and their extra regularity.Central-upwind scheme for a non-hydrostatic Saint-Venant system.https://www.zbmath.org/1460.651052021-06-15T18:09:00+00:00"Chertock, Alina"https://www.zbmath.org/authors/?q=ai:chertock.alina-e"Kurganov, Alexander"https://www.zbmath.org/authors/?q=ai:kurganov.alexander"Miller, Jason"https://www.zbmath.org/authors/?q=ai:miller.jason"Yan, Jun"https://www.zbmath.org/authors/?q=ai:yan.junThe authors develop a second-order central-upwind scheme for a nonhydrostatic version of the Saint-Venant system. This scheme is well-balanced and positivity preserving. The scheme is used to study ability of the non-hydrostatic Saint-Venant system to model long-time propagation and on-shore arrival of the tsunami-type waves. It is remarked that for a certain range of the dispersive coefficients, both the shape and amplitude of the waves are preserved even when the computational grid is coarse. The importance of the dispersive terms in the description of on-shore arrival is shown.
For the entire collection see [Zbl 1453.35003].
Reviewer: Abdallah Bradji (Annaba)New general decay result for a system of viscoelastic wave equations with past history.https://www.zbmath.org/1460.350342021-06-15T18:09:00+00:00"Al-Mahdi, Adel M."https://www.zbmath.org/authors/?q=ai:al-mahdi.adel-m"Al-Gharabli, Mohammad M."https://www.zbmath.org/authors/?q=ai:algharabli.mohammad-m"Messaoudi, Salim A."https://www.zbmath.org/authors/?q=ai:messaoudi.salim-aSummary: This work is concerned with a coupled system of viscoelastic wave equations in the presence of infinite-memory terms. We show that the stability of the system holds for a much larger class of kernels. More precisely, we consider the kernels \(g_i:[0,+\infty)\rightarrow (0,+\infty)\) satisfying
\[
g_i'(t)\leq-\xi_i(t)H_i(g_i(t)),\quad\forall\,t\geq0 \quad\text{and for }i=1,2,
\]
where \(\xi_i\) and \(H_i\) are functions satisfying some specific properties. Under this very general assumption on the behavior of \(g_i\) at infinity, we establish a relation between the decay rate of the solutions and the growth of \(g_i\) at infinity. This work generalizes and improves earlier results in the literature. Moreover, we drop the boundedness assumptions on the history data, usually made in the literature.Sensitivity analysis of Burgers' equation with shocks.https://www.zbmath.org/1460.651292021-06-15T18:09:00+00:00"Li, Qin"https://www.zbmath.org/authors/?q=ai:li.qin"Liu, Jian-Guo"https://www.zbmath.org/authors/?q=ai:liu.jian-guo"Shu, Ruiwen"https://www.zbmath.org/authors/?q=ai:shu.ruiwenSensitivity analysis of Burgers' equation with shocks is discussed. It is known that the generalized polynomial chaos (gPC) method is used in many problems. For gPC to achieve high accuracy, PDE solutions need to have high regularity, but as usual, this is not true for hyperbolic type problems. In this paper, a counterargument is provided and is shown that even though the solution profile develops singularities, which destroys the spectral accuracy of gPC, the physical quantities are all smooth functions of the uncertainties coming from both initial data and the wave speed. The paper is organized as follows. Section 1 is an introduction. In Section 2, some notations are introduced and the precise quantitative versions of the main two theorems are stated. The deterministic case is focused on in Section 3 and some necessary tools for analyzing are prepared too. These tools are used in Section 4 and the abovementioned two main theorems are proved. The results to treat conservation laws with general convex fluxes are extended in Section 5. Conclusions are given in Section 6 and finally, some proofs not essential to the main context are given in the Appendix.
Reviewer: Temur A. Jangveladze (Tbilisi)A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain.https://www.zbmath.org/1460.352342021-06-15T18:09:00+00:00"Fino, Ahmad Z."https://www.zbmath.org/authors/?q=ai:fino.ahmad-z"Chen, Wenhui"https://www.zbmath.org/authors/?q=ai:chen.wenhuiSummary: We study two-dimensional semilinear strongly damped wave equation with mixed nonlinearity \(|u|^p+|u_t|^q\) in an exterior domain, where \(p,q>1\). We prove global (in time) existence of small data solution with suitable higher regularity by using a weighted energy method, and assuming some conditions on powers of nonlinearity.Minimal time for the exact controllability of one-dimensional first-order linear hyperbolic systems by one-sided boundary controls.https://www.zbmath.org/1460.352232021-06-15T18:09:00+00:00"Hu, Long"https://www.zbmath.org/authors/?q=ai:hu.long"Olive, Guillaume"https://www.zbmath.org/authors/?q=ai:olive.guillaumeAuthors' abstract: In this article we study the minimal time for the exact controllability of onedimensional first-order linear hyperbolic systems when all the controls are acting on the same side of the boundary. We establish an explicit and easy-to-compute formula for this time with respect to all the coupling parameters of the system. The proof relies on the introduction of a canonical UL-decomposition and the compactnessuniqueness method.
Reviewer: Kaïs Ammari (Monastir)Wave and Klein-Gordon equations on certain locally symmetric spaces.https://www.zbmath.org/1460.353362021-06-15T18:09:00+00:00"Zhang, Hong-Wei"https://www.zbmath.org/authors/?q=ai:zhang.hongweiSummary: This paper is devoted to study the dispersive properties of the linear Klein-Gordon and wave equations on a class of locally symmetric spaces. As a consequence, we obtain the Strichartz estimate and prove global well-posedness results for the corresponding semilinear equation with low regularity data as on real hyperbolic spaces.Stability of smooth solutions for the compressible Euler equations with time-dependent damping and one-side physical vacuum.https://www.zbmath.org/1460.350312021-06-15T18:09:00+00:00"Pan, Xinghong"https://www.zbmath.org/authors/?q=ai:pan.xinghongThe author considers the 1D compressible Euler equations with time-dependent damping and focuses on stability of the one-side vacuum solutions. The gas is isentropic and adiabatic. The domain is time-dependent, with gas expanding from the initial compact domain. The author proves the global existence and stability of smooth solutions. Actually, the solution is obtained in the explicit form and its stability is established.
Reviewer: Ilya A. Chernov (Petrozavodsk)Nonlinear stability in three-layer channel flows.https://www.zbmath.org/1460.760632021-06-15T18:09:00+00:00"Papaefthymiou, E. S."https://www.zbmath.org/authors/?q=ai:papaefthymiou.e-s"Papageorgiou, D. T."https://www.zbmath.org/authors/?q=ai:papageorgiou.demetrios-tSummary: The nonlinear stability of viscous, immiscible multilayer flows in plane channels driven both by a pressure gradient and gravity is studied. Three fluid phases are present with two interfaces. Weakly nonlinear models of coupled evolution equations for the interfacial positions are derived and studied for inertialess, stably stratified flows in channels at small inclination angles. Interfacial tension is demoted and high-wavenumber stabilisation enters due to density stratification through second-order dissipation terms rather than the fourth-order ones found for strong interfacial tension. An asymptotic analysis is carried out to demonstrate how these models arise. The governing equations are \(2\times 2\) systems of second-order semi-linear parabolic partial differential equations (PDEs) that can exhibit inertialess instabilities due to interaction between the interfaces. Mathematically this takes place due to a transition of the nonlinear flux function from hyperbolic to elliptic behaviour. The concept of hyperbolic invariant regions, found in nonlinear parabolic systems, is used to analyse this inertialess mechanism and to derive a transition criterion to predict the large-time nonlinear state of the system. The criterion is shown to predict nonlinear stability or instability of flows that are stable initially, i.e. the initial nonlinear fluxes are hyperbolic. Stability requires the hyperbolicity to persist at large times, whereas instability sets in when ellipticity is encountered as the system evolves. In the former case the solution decays asymptotically to its uniform base state, while in the latter case nonlinear travelling waves can emerge that could not be predicted by a linear stability analysis. The nonlinear analysis predicts threshold initial disturbances above which instability emerges.On the Riemann problem for a hyperbolic system of temple class.https://www.zbmath.org/1460.352292021-06-15T18:09:00+00:00"Guerrero, Richard A. De La Cruz"https://www.zbmath.org/authors/?q=ai:guerrero.richard-a-de-la-cruz"Juajibioy, Juan C."https://www.zbmath.org/authors/?q=ai:juajibioy.juan-carlosSummary: In this chapter, we study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of temple class. Under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of particular delta-shock type solutions are established. Moreover, we show explicitly the solution of generalized Riemann problem.
For the entire collection see [Zbl 1314.49001].Existence and regularity results for terminal value problem for nonlinear fractional wave equations.https://www.zbmath.org/1460.353682021-06-15T18:09:00+00:00"Bao, Ngoc Tran"https://www.zbmath.org/authors/?q=ai:bao.ngoc-tran"Caraballo, Tomás"https://www.zbmath.org/authors/?q=ai:caraballo.tomas"Tuan, Nguyen Huy"https://www.zbmath.org/authors/?q=ai:nguyen-huy-tuan."Zhou, Yong"https://www.zbmath.org/authors/?q=ai:zhou.yong.1Small data global regularity for the 3-D Ericksen-Leslie hyperbolic liquid crystal model without kinematic transport.https://www.zbmath.org/1460.352872021-06-15T18:09:00+00:00"Huang, Jiaxi"https://www.zbmath.org/authors/?q=ai:huang.jiaxi"Jiang, Ning"https://www.zbmath.org/authors/?q=ai:jiang.ning"Luo, Yi-Long"https://www.zbmath.org/authors/?q=ai:luo.yi-long"Zhao, Lifeng"https://www.zbmath.org/authors/?q=ai:zhao.lifengThis work considers the hyperbolic Ericksen-Leslie system, which combined the hydrodynamical equation of motion with a constitutive equation for the orientation field that, in complex, models the motion of liquid crystals. The main result reported in prooving the existence of a unique global solution for such a system satisfying the energy bounds, which are provided.
Reviewer: Eugene Postnikov (Kursk)Finite time blow-up for a nonlinear viscoelastic Petrovsky equation with high initial energy.https://www.zbmath.org/1460.352322021-06-15T18:09:00+00:00"Liu, Lishan"https://www.zbmath.org/authors/?q=ai:liu.lishan"Sun, Fenglong"https://www.zbmath.org/authors/?q=ai:sun.fenglong"Wu, Yonghong"https://www.zbmath.org/authors/?q=ai:wu.yonghong.1The authors study the integro-differential initial boundary value problem
\begin{align*}
&u_{tt}+\Delta^2 u - \int_0^t g(t-\tau)\Delta^2 u(\tau)d\tau + u_t =|u|^{p-2}u\ \text{in}\ \Omega\times (0,T),\\
&u(x,t)-\partial_\nu u(x,t) = 0\ \text{on}\ \partial\Omega\times (0,T),\quad u(x,0)=u_0(x),\ u_t(x,0) = u_1(x)\ \text{in}\ \Omega,
\end{align*}
where \(\ p>2,\ \Omega\subset \mathbb{R}^n,\ n\ge 1,\ g:\mathbb{R}^+\to \mathbb{R}^+\) is nonincreasing, \(\int_0^\infty g(s)ds<1, \ (u_0,u_1)\in H_0^2\times L^2(\Omega).\) The local existence theorem and the energy identity are mentioned without any proof. The main result is the blow up in a finite time \(T^*\) (the maximal existence time) of the weak solution \(u\).
Reviewer: Igor Bock (Bratislava)Flux-approximation limits of solutions to the brio system with two independent parameters.https://www.zbmath.org/1460.352302021-06-15T18:09:00+00:00"Zhang, Yanyan"https://www.zbmath.org/authors/?q=ai:zhang.yanyan"Zhang, Yu"https://www.zbmath.org/authors/?q=ai:zhang.yu.3Summary: By the flux-approximation method, we study limits of Riemann solutions to the Brio system with two independent parameters. The Riemann problem of the perturbed system is solved analytically, and four kinds of solutions are obtained constructively. It is shown that, as the two-parameter flux perturbation vanishes, any two-shock-wave and two-rarefaction-wave solutions of the perturbed Brio system converge to the delta-shock and vacuum solutions of the transport equations, respectively. In addition, we specially pay attention to the Riemann problem of a perturbed simplified system of conservation laws derived from the perturbed Brio system by neglecting some quadratic term. As one of the parameters of the perturbed Brio system goes to zero, the solution of which consisting of two shock waves tends to a delta-shock solution to this simplified system. By contrast, the solution containing two rarefaction waves converges to a contact discontinuity and a rarefaction wave of the simplified system. What is more, the formation mechanisms of delta shock waves under flux approximation with both two parameters and only one parameter are clarified. Some numerical simulations presenting the formation processes of delta shock waves and vacuum states are also presented to confirm the theory analysis.Motion of interfaces for a damped hyperbolic Allen-Cahn equation.https://www.zbmath.org/1460.350132021-06-15T18:09:00+00:00"Folino, Raffaele"https://www.zbmath.org/authors/?q=ai:folino.raffaele"Lattanzio, Corrado"https://www.zbmath.org/authors/?q=ai:lattanzio.corrado"Mascia, Corrado"https://www.zbmath.org/authors/?q=ai:mascia.corradoSummary: This paper concerns with the motion of the interface for a damped hyperbolic Allen-Cahn equation, in a bounded domain of \(\mathbb{R}^n\), for \(n=2\) or \(n=3\). In particular, we focus the attention on radially symmetric solutions and extend to the hyperbolic framework some well-known results of the classic parabolic case: it is shown that, under appropriate assumptions on the initial data and on the boundary conditions, the interface moves by mean curvature as the diffusion coefficient goes to \(0\).A posteriori error estimates for self-similar solutions to the Euler equations.https://www.zbmath.org/1460.352682021-06-15T18:09:00+00:00"Bressan, Alberto"https://www.zbmath.org/authors/?q=ai:bressan.alberto"Shen, Wen"https://www.zbmath.org/authors/?q=ai:shen.wenSummary: The main goal of this paper is to analyze a family of ``simplest possible'' initial data for which, as shown by numerical simulations, the incompressible Euler equations have multiple solutions. We take here a first step toward a rigorous validation of these numerical results. Namely, we consider the system of equations corresponding to a self-similar solution, restricted to a bounded domain with smooth boundary. Given an approximate solution obtained via a finite dimensional Galerkin method, we establish a posteriori error bounds on the distance between the numerical approximation and the exact solution having the same boundary data.On averaged exponential integrators for semilinear wave equations with solutions of low-regularity.https://www.zbmath.org/1460.650552021-06-15T18:09:00+00:00"Buchholz, Simone"https://www.zbmath.org/authors/?q=ai:buchholz.simone"Dörich, Benjamin"https://www.zbmath.org/authors/?q=ai:dorich.benjamin"Hochbruck, Marlis"https://www.zbmath.org/authors/?q=ai:hochbruck.marlisSummary: In this paper we introduce a class of second-order exponential schemes for the time integration of semilinear wave equations. They are constructed such that the established error bounds only depend on quantities obtained from a well-posedness result of a classical solution. To compensate missing regularity of the solution the proofs become considerably more involved compared to a standard error analysis. Key tools are appropriate filter functions as well as the integration-by-parts and summation-by-parts formulas. We include numerical examples to illustrate the advantage of the proposed methods.A fractional degenerate parabolic-hyperbolic Cauchy problem with noise.https://www.zbmath.org/1460.353692021-06-15T18:09:00+00:00"Bhauryal, Neeraj"https://www.zbmath.org/authors/?q=ai:bhauryal.neeraj"Koley, Ujjwal"https://www.zbmath.org/authors/?q=ai:koley.ujjwal"Vallet, Guy"https://www.zbmath.org/authors/?q=ai:vallet.guySummary: We consider the Cauchy problem for a stochastic scalar parabolic-hyperbolic equation in any space dimension with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators. We adapt the notion of stochastic entropy solution and provide a new technical framework to prove the uniqueness. The existence proof relies on the vanishing viscosity method. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities and derive error estimate for the stochastic vanishing viscosity method. In addition, we develop uniqueness method ``à la Kružkov'' for more general equations where the noise coefficient may depend explicitly on the spatial variable.Semi-algebraic sets method in PDE and mathematical physics.https://www.zbmath.org/1460.350112021-06-15T18:09:00+00:00"Wang, W.-M."https://www.zbmath.org/authors/?q=ai:wang.weimin|wang.wenming|wang.wumin|wang.whei-ming|wang.wei-min|wang.weiming|wang.wenminSummary: This paper surveys recent progress in the analysis of nonlinear partial differential equations using Anderson localization and semi-algebraic sets method. We discuss the application of these tools from linear analysis to nonlinear equations such as the nonlinear Schrödinger equations, the nonlinear Klein-Gordon equations (nonlinear wave equations), and the nonlinear random Schrödinger equations on the lattice. We also review the related linear time-dependent problems.
{\copyright 2021 American Institute of Physics}Chaotic behaviors of one-dimensional wave equations with van der Pol boundary conditions containing a source term.https://www.zbmath.org/1460.352222021-06-15T18:09:00+00:00"Chen, Zhijing"https://www.zbmath.org/authors/?q=ai:chen.zhijing"Huang, Yu"https://www.zbmath.org/authors/?q=ai:huang.yu"Sun, Haiwei"https://www.zbmath.org/authors/?q=ai:sun.haiwei"Zhou, Tongyang"https://www.zbmath.org/authors/?q=ai:zhou.tongyang